Subgroups — #34260261

Order 20224.

572 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773983	C₂	1	1	yes	maximal, characteristic
2	#23772498	C₂	1	2	yes	maximal
3	#23762296	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762269	C₂	1	2	yes	maximal, complemented
5	#23762268	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762147	C₂	1	1	yes	maximal, characteristic
9	#23761819	C₂	1	2	yes	maximal
10	C₈×#11929607	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099117	C₂²	1	2	yes	—
13	#19098967	C₄	1	1	yes	characteristic
14	#19098941	C₄	1	1	yes	characteristic
15	#19098912	C₄	1	2	yes	—
16	#19098805	C₄	1	1	yes	characteristic
17	#19098803	C₄	1	2	yes	—
18	#19098801	C₄	1	1	yes	characteristic
19	#19098691	—	2	2	no	—
20	#19098690	C₂²	1	2	yes	—
21	Q₈×C₇₉⋊C₈	C₂²	1	2	yes	—
22	C₈×Dic₁₅₈	C₂²	1	2	yes	—
23	#19098687	—	2	2	no	—
24	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
25	C₄×#11929614	C₂²	1	2	yes	—
26	#19098625	—	2	2	no	—
27	#19098623	—	2	2	no	—
28	#19098615	C₂²	1	2	yes	—
29	#19098614	C₂²	1	1	yes	characteristic
30	C₄×#11929607	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
32	#19098316	C₂²	1	1	yes	characteristic
33	#19098313	C₂²	1	2	yes	—
34	#19098295	C₂²	1	2	yes	—
35	Q₈⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
36	#19098293	C₂²	1	2	yes	—
37	#19098285	C₂²	1	1	yes	characteristic
38	#19098284	C₂²	1	1	yes	characteristic
39	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₇₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	#335×C₇₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₉	C₂²	1	2	yes	—
44	C₈²×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11929614	C₂×C₄	1	2	yes	—
47	C₂×#11929607	—	2	1	no	—
48	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
50	#14579160	C₂×C₄	1	2	yes	—
51	#14579159	C₂×C₄	1	2	yes	—
52	#14579151	—	2	2	no	—
53	#14579141	C₂×C₄	1	2	yes	—
54	#14579139	—	2	2	no	—
55	#14579137	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₅₈	—	2	2	no	—
57	C₄×Dic₁₅₈	C₂³	1	2	yes	—
58	#14579118	—	2	2	no	—
59	C₄²×D₇₉	C₂³	1	1	yes	characteristic
60	#14579095	C₂×C₄	1	1	yes	characteristic
61	#14579094	C₂×C₄	1	2	yes	—
62	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
63	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
64	Dic₇₉⋊C₈	—	2	2	no	—
65	Dic₇₉⋊C₈	—	2	2	no	—
66	Dic₇₉⋊C₈	D₄	1	1	yes	characteristic
67	Dic₇₉⋊C₈	D₄	1	1	yes	characteristic
68	#14579086	C₂×C₄	1	2	yes	—
69	#14579085	C₂×C₄	1	2	yes	—
70	#14579084	C₂×C₄	1	2	yes	—
71	(C₂×D₇₉)⋊C₈	—	2	2	no	—
72	(C₂×D₇₉)⋊C₈	D₄	1	1	yes	characteristic
73	(C₂×D₇₉)⋊C₈	D₄	1	1	yes	characteristic
74	#14579071	—	2	2	no	—
75	#14579071	—	2	1	no	—
76	C₄×C₇₉⋊C₈	—	2	2	no	—
77	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×Dic₇₉	—	2	2	no	—
81	C₈×Dic₇₉	—	2	2	no	—
82	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₇₉	C₂³	1	2	yes	—
86	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₇₉	—	2	2	no	—
88	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
89	#103×C₇₉	C₂×C₄	1	2	yes	—
90	#97×C₇₉	—	2	2	no	—
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₅₈	—	2	2	no	—
95	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
99	Dic₃₁₆	—	2	2	no	—
100	#11929614	—	4	2	no	—
101	#11929607	—	2	4	no	—
102	C₂×C₇₉⋊C₈	—	2	2	no	—
103	C₂×C₇₉⋊C₈	—	2	1	no	—
104	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₈×D₇₉	—	2	2	no	—
109	C₈×D₇₉	—	4	1	no	—
110	C₄⋊Dic₇₉	—	2	2	no	—
111	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
112	Dic₇₉⋊C₄	—	2	2	no	—
113	Dic₇₉⋊C₄	—	2	2	no	—
114	Dic₇₉⋊C₄	—	2	2	no	—
115	Dic₇₉⋊C₄	—	2	2	no	—
116	Dic₇₉⋊C₄	—	2	2	no	—
117	(C₂×D₇₉)⋊C₄	—	2	2	no	—
118	(C₂×D₇₉)⋊C₄	—	2	2	no	—
119	C₄×Dic₇₉	—	2	2	no	—
120	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₇₉	#41	1	1	yes	characteristic
123	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
124	Q₁₆×C₇₉	—	2	2	no	—
125	C₂×C₈×C₇₉	—	2	2	no	—
126	C₂×C₈×C₇₉	—	2	1	no	—
127	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₇₉	—	2	2	no	—
132	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
133	C₄²×C₇₉	—	2	2	no	—
134	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
135	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₇₉	—	2	2	no	—
137	Q₈×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	—	2	2	no	—
139	C₂×C₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	2	no	—
145	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
147	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
148	C₄×D₇₉	—	2	1	no	—
149	C₄×D₇₉	—	2	1	no	—
150	C₄×D₇₉	—	2	1	no	—
151	C₄×D₇₉	—	2	1	no	—
152	C₄×D₇₉	—	2	2	no	—
153	Dic₁₅₈	—	2	2	no	—
154	Dic₁₅₈	—	2	2	no	—
155	Dic₁₅₈	—	4	4	no	—
156	C₈×C₇₉	—	2	2	no	—
157	C₈×C₇₉	—	2	1	no	—
158	C₈×C₇₉	—	2	1	no	—
159	C₈×C₇₉	—	2	1	no	—
160	C₈×C₇₉	—	4	2	no	—
161	C₈×C₇₉	#131	1	2	yes	—
162	C₇₉⋊C₈	—	2	2	no	—
163	C₇₉⋊C₈	—	2	1	no	—
164	C₇₉⋊C₈	—	2	1	no	—
165	C₇₉⋊C₈	—	2	1	no	—
166	C₇₉⋊C₈	—	2	1	no	—
167	C₇₉⋊C₈	—	4	2	no	—
168	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×D₇₉	—	2	1	no	—
170	C₂×D₇₉	—	2	1	no	—
171	C₂×D₇₉	—	2	1	no	—
172	C₄×C₇₉	—	2	1	no	—
173	C₄×C₇₉	—	4	2	no	—
174	C₄×C₇₉	—	4	2	no	—
175	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
176	C₄×C₇₉	#568	1	1	yes	characteristic
177	C₄×C₇₉	#578	1	1	yes	characteristic
178	C₄×C₇₉	#578	1	1	yes	characteristic
179	Dic₇₉	—	2	1	no	—
180	Dic₇₉	—	2	1	no	—
181	Dic₇₉	—	4	2	no	—
182	Dic₇₉	—	4	2	no	—
183	Dic₇₉	#443	1	2	yes	—
184	#18086	—	79	1	no	maximal, Sylow
185	C₂×C₇₉	#2935	1	1	yes	characteristic
186	C₂×C₇₉	#2950	1	1	yes	characteristic
187	C₂×C₇₉	C₂×#443	1	1	yes	characteristic
188	D₇₉	—	4	1	no	—
189	#3170	—	79	1	no	—
190	#2986	—	79	2	no	—
191	#1591	—	79	1	no	—
192	#1591	D₇₉	1	1	yes	characteristic, complemented
193	#1587	—	79	2	no	—
194	#1587	—	79	2	no	—
195	C₈×Q₁₆	—	79	1	no	—
196	C₈×Q₁₆	—	79	1	no	—
197	#1564	—	79	1	no	—
198	#1485	—	79	2	no	—
199	C₈×M₁₆	—	79	1	no	—
200	C₇₉	#18086	1	1	yes	characteristic, complemented, Sylow
201	#533	—	79	2	no	—
202	#500	—	79	1	no	—
203	#494	—	79	1	no	—
204	#487	—	79	2	no	—
205	#467	—	79	1	no	—
206	#467	—	79	1	no	—
207	#467	—	79	2	no	—
208	#446	—	79	2	no	—
209	#446	—	158	2	no	—
210	#446	C₂×D₇₉	1	2	yes	—
211	C₈×Q₈	—	79	2	no	—
212	C₈×Q₈	—	79	2	no	—
213	C₈×Q₈	—	158	2	no	—
214	C₄×Q₁₆	—	79	2	no	—
215	C₄×Q₁₆	—	79	1	no	—
216	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
217	#433	—	158	2	no	—
218	#433	—	158	2	no	—
219	#431	—	79	1	no	—
220	#431	—	79	2	no	—
221	C₄×M₁₆	—	79	1	no	—
222	C₂×C₄×C₈	—	79	1	no	—
223	#335	—	79	1	no	—
224	#335	C₂×D₇₉	1	1	yes	characteristic
225	#334	—	79	2	no	—
226	Q₈⋊C₈	—	79	2	no	—
227	Q₈⋊C₈	—	79	2	no	—
228	Q₈⋊C₈	—	79	2	no	—
229	Q₈⋊C₈	C₂×D₇₉	1	2	yes	—
230	#322	—	79	1	no	—
231	#322	—	79	1	no	—
232	C₈²	—	79	1	no	—
233	C₈²	C₂×D₇₉	1	1	yes	characteristic
234	C₂×Q₁₆	—	79	2	no	—
235	C₂×Q₁₆	—	79	1	no	—
236	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
237	C₂×M₁₆	—	158	1	no	—
238	C₂²×C₈	—	79	1	no	—
239	C₂²×C₈	—	79	1	no	—
240	C₄⋊Q₈	—	79	2	no	—
241	C₄⋊Q₈	—	79	2	no	—
242	#125	—	158	2	no	—
243	C₂²⋊Q₈	—	79	2	no	—
244	C₂²⋊Q₈	—	79	2	no	—
245	C₂²⋊Q₈	—	158	2	no	—
246	C₄×Q₈	—	79	2	no	—
247	C₄×Q₈	—	158	2	no	—
248	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
249	#117	—	158	2	no	—
250	C₂×C₄²	—	79	1	no	—
251	#107	—	79	1	no	—
252	#107	—	79	2	no	—
253	#107	C₄×D₇₉	1	1	yes	characteristic
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	79	1	no	—
256	C₄⋊C₈	—	79	1	no	—
257	C₄⋊C₈	—	79	2	no	—
258	C₄⋊C₈	—	158	2	no	—
259	C₄⋊C₈	—	158	2	no	—
260	C₄⋊C₈	—	158	2	no	—
261	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
262	#103	—	79	2	no	—
263	#103	—	79	2	no	—
264	#103	—	79	2	no	—
265	#103	C₄×D₇₉	1	2	yes	—
266	C₂²⋊C₈	—	79	1	no	—
267	C₂²⋊C₈	—	79	1	no	—
268	C₂²⋊C₈	—	158	2	no	—
269	#97	—	2	2	no	—
270	#97	—	158	1	no	—
271	#97	—	158	2	no	—
272	C₄×C₈	—	79	1	no	—
273	C₄×C₈	—	79	1	no	—
274	C₄×C₈	—	79	1	no	—
275	C₄×C₈	—	79	1	no	—
276	C₄×C₈	—	79	1	no	—
277	C₄×C₈	—	158	2	no	—
278	C₄×C₈	—	158	2	no	—
279	C₄×C₈	—	158	2	no	—
280	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
283	C₂×Q₈	—	79	2	no	—
284	C₂×Q₈	—	158	2	no	—
285	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
286	C₂²×C₄	—	79	1	no	—
287	C₂²×C₄	—	79	1	no	—
288	C₂²×C₄	—	79	1	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	316	2	no	—
291	Q₁₆	—	158	2	no	—
292	M₁₆	—	158	4	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	316	1	no	—
296	C₂×C₈	—	79	1	no	—
297	C₂×C₈	—	79	1	no	—
298	C₂×C₈	—	79	1	no	—
299	C₂×C₈	—	79	1	no	—
300	C₂×C₈	—	158	2	no	—
301	C₂×C₈	—	158	1	no	—
302	C₂×C₈	—	158	2	no	—
303	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
305	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
306	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	79	2	no	—
309	C₄⋊C₄	—	158	2	no	—
310	C₄⋊C₄	—	158	2	no	—
311	C₄⋊C₄	—	158	2	no	—
312	C₄⋊C₄	—	158	2	no	—
313	C₄⋊C₄	—	158	2	no	—
314	C₄⋊C₄	—	158	2	no	—
315	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
316	C₂²⋊C₄	—	158	2	no	—
317	C₂²⋊C₄	—	158	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	79	1	no	—
320	C₄²	—	79	1	no	—
321	C₄²	—	79	1	no	—
322	C₄²	—	158	2	no	—
323	C₄²	C₂³×D₇₉	1	1	yes	characteristic
324	C₂³	—	79	1	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	Q₈	—	316	4	no	—
328	Q₈	—	158	2	no	—
329	Q₈	—	158	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	79	1	no	—
333	C₂×C₄	—	79	1	no	—
334	C₂×C₄	—	79	1	no	—
335	C₂×C₄	—	158	2	no	—
336	C₂×C₄	—	158	2	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	—	158	1	no	—
339	C₂×C₄	—	158	1	no	—
340	C₂×C₄	—	158	1	no	—
341	C₂×C₄	—	158	1	no	—
342	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	4	2	no	—
350	C₈	—	316	2	no	—
351	C₈	—	158	2	no	—
352	C₈	—	158	1	no	—
353	C₈	—	158	1	no	—
354	C₈	—	158	1	no	—
355	C₈	—	158	1	no	—
356	C₈	#14579169	1	2	yes	—
357	C₂²	—	158	1	no	—
358	C₂²	—	158	1	no	—
359	C₂²	—	158	1	no	—
360	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	316	2	no	—
365	C₄	—	316	2	no	—
366	C₄	—	79	2	no	—
367	C₄	—	158	1	no	—
368	C₄	—	158	1	no	—
369	C₄	C₂×#14579169	1	1	yes	characteristic
370	C₄	#19099328	1	1	yes	characteristic
371	C₄	#19099376	1	1	yes	characteristic
372	C₄	#19099377	1	1	yes	characteristic
373	C₂	—	316	1	no	—
374	C₂	#23772145	1	1	yes	characteristic
375	C₂	#23772263	1	1	yes	characteristic
376	C₂	#443×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1591	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂²×D₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1591	D₇₉
hypocenter	C₇₉	#18086
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #23762296	C₂	semidirect
4. #23762269	C₂	semidirect
5. #23762268	C₂	semidirect
6. C₈×Dic₃₁₆	C₂	semidirect
7. Q₁₆×C₇₉⋊C₈	C₂	semidirect
11. C₇₉×#1591	C₂	semidirect
192. #1591	D₇₉	semidirect
200. C₇₉	#18086	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23773983	2
#23772498	2
#23762296	2
#23762269	2
#23762268	2
C₈×Dic₃₁₆	2
Q₁₆×C₇₉⋊C₈	2
#23762147	2
#23761819	2
C₈×#11929607	2
C₇₉×#1591	2
#18086	79